Local and Global Casimir Energies: Divergences, Renormalization, and the
Coupling to Gravity

Abstract

From the beginning of the subject, calculations of quantum vacuum
energies or Casimir
energies have been plagued with two types of divergences: The total energy,
which may be thought of as some sort of
regularization of the zero-point energy, ∑12ℏω, seems
manifestly divergent. And local energy densities, obtained from the vacuum
expectation
value of the energy-momentum tensor, ⟨T00⟩, typically
diverge near
boundaries. These two types of divergences have little to do with each other.
The energy of interaction between distinct rigid bodies of whatever type is
finite, corresponding to observable forces and torques between the bodies,
which can be unambiguously calculated. The divergent local energy densities
near surfaces do not change when the relative position of the rigid bodies is
altered. The self-energy
of a body is less well-defined, and suffers divergences which may or may not
be removable.
Some examples where a unique total self-stress may be evaluated include the
perfectly conducting
spherical shell first considered by Boyer, a perfectly conducting cylindrical
shell, and
dilute dielectric balls and cylinders. In these cases the finite part
is unique, yet
there are divergent contributions which may be subsumed in some sort of
renormalization
of physical parameters. The finiteness of self-energies is separate from the
issue of
the physical observability of the effect. The divergences that occur
in the local
energy-momentum tensor near surfaces are distinct from the divergences in the
total
energy, which are often associated with energy located
exactly on the surfaces.
However, the local energy-momentum tensor couples to gravity, so what
is the significance of infinite quantities here? For the classic situation
of parallel plates there are indications that the divergences in the local
energy density are consistent with divergences in Einstein’s equations;
correspondingly, it has been shown that divergences in the total Casimir
energy serve to precisely renormalize
the masses of the plates, in accordance with the equivalence principle. This
should be a general property, but has not yet been established, for
example, for the Boyer sphere. It is known that such local divergences
can have no effect on macroscopic causality.

For more than 60 years it has been appreciated that quantum fluctuations
can give rise to macroscopic forces between bodies casimir ().
These can be thought
of as the sum, in general nonlinear, of the van der Waals forces between the
constituents of the bodies, which, in the 1930s had been shown by London
london ()
to arise from dipole-dipole interactions in the nonretarded regime, and in
1947 to arise from the same interactions in the retarded regime, giving rise
to so-called Casimir-Polder forces
casimirandpolder (). Bohr casimir50 ()
apparently provided the incentive
to Casimir to rederive the macroscopic force between a molecule and a
surface, and then derive the force between two conducting surfaces, directly
in terms of zero-point fluctuations of the electromagnetic fields in which
the bodies are immersed. But these two points of view—action at a distance
and local action—are essentially equivalent, and one implies the other,
not withstanding some objections to the latter Jaffe:2003ji ().

The quantum-vacuum-fluctuation force between two parallel surfaces—be they
conductors or dielectrics lifshitz (); dzyaloshinskii0 (); dzyaloshinskii ()
—was the first situation considered, and still
the only one accessible experimentally.
(For a current review of the experimental situation, see
Bordag:2009zz (); Klimchitskaya:2009cw ())
Actually, most experiments measure
the force between a spherical surface and a plane, but the surfaces are so
close together that the force may be obtained from the parallel plate case
by a geometrical transformation, the so-called proximity force approximation
(PFA) derpt (); derpt2 (); blocki ().
However, it is not possible to find an extension to the PFA beyond
the first approximation of the separation distance being smaller than all
other scales in the problem.
In the last few years, advances in technique have allowed quasi-analytical
and numerical
calculations to be carried out between bodies of essentially any shape, at
least at medium to large separation, so the limitations of the PFA may be
largely transcended. (For the current status of these developments,
see the contributions to this volume by Emig, Jaffe, and Rahi, and by
Johnson; for earlier references, see, for example Milton:2008st ().)
These advances have shifted calculational attention
away from what used to be the central challenge in Casimir theory, how to
define and calculate Casimir energies and self-stresses of single bodies.

There are, of course, sound reasons for this. Forces between distinct
bodies are necessarily physically finite, and can, and have, been observed
by experiment. Self-energies or self-stresses typically involve divergent
quantities which are difficult to remove, and have obscure physical meaning.
For example, the self-stress on a perfectly conducting spherical shell of
negligible thickness was calculated by Boyer in 1968 Boyer:1968uf (),
who found a repulsive
self-stress that has subsequently been confirmed by a variety of techniques.
Yet it remains unclear what physical significance this energy has. If
the sphere is bisected and the two halves pulled apart, there will be
an attraction (due to the closest parts of the hemispheres) not a repulsion.
The same remarks, although exacerbated, apply to the self-stress on a
rectangular box lukosz (); lukosz1 (); lukosz2 (); ambjorn ().
The situation in that case is worse because (1) the
sharp corners give rise to additional divergences not present in the
case of a smooth boundary (it has been proven that the self-energy of
a smooth closed infinitesimally thin conducting surface is finite
balian (); Bernasconi ()), and (2) the
exterior contributions cannot be computed because the vector Helmholtz
equation cannot be separated. But calculational challenges aside,
the physical significance of self-energy remains elusive.

The exception to this objection is provided by gravity. Gravity
couples to the local energy-momentum or stress tensor, and, in the
leading quantum approximation, it is the vacuum expectation value
of the stress tensor that provides the source term in Einstein’s equations.
Self energies should therefore in principle be observable. This
is largely uncharted territory, except in the instance of the classic
situation of parallel plates. There, after a bit of initial confusion,
it has now been established that the divergent self-energies of each
plate in a two-plate apparatus, as well as the mutual Casimir energy
due to both plates, gravitates according to the equivalence principle,
so that indeed it is consistent to absorb the divergent self-energies of
each plate into the gravitational and inertial mass of each
Fulling:2007xa (); Milton:2007ar ().
This should be a universal feature.

In this paper, for pedagogical reasons, we will concentrate attention
on the Casimir effect due to massless scalar field fluctuations, where
the potentials are described by δ-function potentials, so-called
semitransparent boundaries. In the limit as the coupling to these potentials
becomes infinitely strong, this imposes Dirichlet boundary conditions.
At least in some cases, Neumann boundary conditions can be achieved
by the strong coupling limit of the derivative of δ-function
potentials. So we can, for planes, spheres, and circular cylinders,
recover in this way the results for electromagnetic field fluctuations
imposed by perfectly conducting boundaries. Since the mutual interaction
between distinct semitransparent bodies have been described in detail
elsewhere Milton:2007gy (); Milton:2007wz (); Wagner:2008qq (), we will,
as implied above, concentrate on the self-interaction issues.

A summary of what is known for spheres and circular cylinders is
given in Table 1.

Table 1: Casimir energy (E) for a sphere and Casimir energy per unit
length (E) for a cylinder, both of radius a.
Here the different boundary
conditions are perfectly conducting for electromagnetic fields (EM),
Dirichlet for scalar fields (D), dilute dielectric for electromagnetic
fields [coefficient of (ε−1)2], dilute dielectric for
electromagnetic fields
with media having the same speed of light (coefficient of ξ2=[(ε−1)/(ε+1)]2),
perfectly conducting surface with eccentricity δe
(coefficient of δe2), and weak coupling
for scalar field with δ-function boundary given by (60),
(coefficient of λ2/a2). The references given are, to the author’s
knowledge, the
first paper in which the results in the various cases were found.

In this section, we will rederive the classic Casimir result for the
force between parallel conducting plates casimir (). Since the
usual Green’s function derivation may be found in monographs miltonbook (),
and was for example reviewed in connection with current controversies over
finiteness of Casimir energies Milton:2002vm (), we will here present
a different approach, based on δ-function potentials, which in the
limit of strong coupling reduce to the appropriate Dirichlet or Robin
boundary conditions of a perfectly conducting surface, as appropriate to
TE and TM modes, respectively. Such potentials were first considered
by the Leipzig group hennig (); bkv (), but more recently have been the focus
of the program of the MIT group graham (); graham2 (); Graham:2002fw (); Graham:2003ib ().
The discussion here is based on a paper by the author
Milton:2004vy (). (See also Milton:2004ya ().)
(A multiple scattering approach to this problem has also
been given in Milton:2007wz ().)

We consider a massive scalar field (mass μ)
interacting with two δ-function
potentials, one at x=0 and one at x=a, which has an interaction
Lagrange density

Lint=−12λδ(x)ϕ2(x)−12λ′δ(x−a)ϕ2(x),

(1)

where the positive coupling constants λ and λ′
have dimensions of mass. In the limit as both
couplings become infinite, these potentials enforce Dirichlet boundary
conditions at the two points:

λ,λ′→∞:ϕ(0),ϕ(a)→0.

(2)

The Casimir energy for this
situation may be computed in terms of the Green’s function G,

G(x,x′)=\I⟨Tϕ(x)ϕ(x′)⟩,

(3)

which has a time Fourier transform,

G(x,x′)=∫\Dω2π\E−\Iω(t−t′)G(x,x′;ω).

(4)

Actually, this is a somewhat symbolic expression, for the Feynman Green’s
function (3) implies that the frequency contour of integration
here must pass below the singularities in ω on the negative real
axis, and above those on the positive real axis kantowski (); Brevik:2000hk ().
Because we have translational invariance in the two directions parallel
to the plates, we have a Fourier transform in those directions as well:

for both fields inside, 0<x,x′<a, while if both field points are outside,
a<x,x′,

g(x,x′)

=

12κ\E−κ|x−x′|+12κΔ\E−κ(x+x′−2a)

(7b)

×[−λ2κ(1−λ′2κ)−λ′2κ(1+λ2κ)\E2κa].

For x,x′<0,

g(x,x′)

=

12κ\E−κ|x−x′|+12κΔ\Eκ(x+x′)

(7c)

×[−λ′2κ(1−λ2κ)−λ2κ(1+λ′2κ)\E2κa].

Here, the denominator is

Δ=(1+λ2κ)(1+λ′2κ)\E2κa−λλ′(2κ)2.

(8)

Note that in the strong coupling limit we recover the familiar results,
for example, inside

λ,λ′→∞:g(x,x′)→−sinhκx<sinhκ(x>−a)κsinhκa.

(9)

Here x>, x< denote the greater, lesser, of x,x′.
Evidently, this Green’s function vanishes at x=0 and at x=a.

Let us henceforward consider μ=0, since otherwise there are
no long-range forces. (There is no nonrelativistic Casimir effect.)
We can now calculate the force on one of the δ-function plates by
calculating the discontinuity of the stress tensor, obtained from the
Green’s function (3) by

⟨Tμν⟩=(∂μ∂ν′−12gμν∂λ∂′λ)1\IG(x,x′)∣∣∣x=x′.

(10)

Writing a reduced stress tensor by

⟨Tμν⟩=∫\Dω2π∫(\Dk)(2π)2tμν,

(11)

we find inside, just to the left of the plate at x=a,

txx∣∣x=a−

=

12\I(−κ2+∂x∂x′)g(x,x′)∣∣∣x=x′=a−

(12a)

=

−κ2\I{1+2λλ′(2κ)21Δ}.

(12b)

From this we must subtract the stress just to the right of the plate at
x=a, obtained from (7b), which turns out to be in the massless
limit

txx∣∣x=a+=−κ2\I,

(13)

which just cancels the 1 in braces in (12b).
Thus the pressure on the plate at x=a due to the quantum fluctuations
in the scalar field is given by the simple, finite expression

while for large λ it approaches half of Casimir’s result casimir ()
for perfectly conducting parallel plates,

PTE∼−π2480a4,λ≫1.

(15b)

We can also compute the energy density. Integrating the energy density
over all space should give rise to the total energy.
Indeed, the above result may be easily
derived from the following expression for the total energy,

E=∫(\Dr)⟨T00⟩

=

12\I∫(\Dr)(∂0∂′0−∇2)G(x,x′)∣∣∣x=x′

(16)

=

12\I∫(\Dr)∫\Dω2π2ω2G(r,r),

if we integrate by parts and omit the surface term.
Integrating over the Green’s functions in the three regions,
given by (7a), (7b), and (7c), we obtain for
λ=λ′,

E=148π2a3∫∞0\Dyy211+y/(λa)−196π2a3∫∞0\Dyy31+2/(y+λa)(y/(λa)+1)2\Ey−1,

(17)

where the first term is regarded as an irrelevant constant (λ is
constant so the a can be scaled out),
and the second term
coincides with the massless limit of the energy first found by Bordag
et al. hennig (), and given in Graham:2003ib (); Weigel:2003tp ().
When differentiated with respect to a,
(17), with λ fixed, yields the pressure
(14). (We will see below that the divergent constant
describe the self-energies of the two plates.)

If, however, we integrate the interior and exterior energy density
directly, one gets a different result.
The origin of this discrepancy with the naive energy
is the existence of a surface contribution
to the energy. To see this, we must include the potential in the stress
tensor,

Tμν=∂μϕ∂νϕ−12gμν(∂λϕ∂λϕ+Vϕ2),

(18)

and then, using the equation of motion, it is immediate to see
that the energy density is

T00=12∂0ϕ∂0ϕ−12ϕ(∂0)2ϕ+12\boldmath{∇}⋅(ϕ\boldmath{∇}ϕ),

(19)

so, because the first two terms here yield the last form in
(16), we conclude that there is
an additional contribution to the energy,

^E

=

−12\I∫\DS⋅\boldmath{∇}G(x,x′)∣∣∣x′=x

(20a)

=

−12\I∫∞−∞\Dω2π∫(\Dk)(2π)2∑\D\Dxg(x,x′)∣∣∣x′=x,

(20b)

where the derivative is taken at the boundaries (here x=0, a) in the
sense of the outward normal from the region in question. When this surface
term is taken into account the extra terms incorporated in (17)
are supplied. The integrated formula (16)
automatically builds in this
surface contribution, as the implicit surface term in the integration
by parts. That is,

E=∫(\Dr)⟨T00⟩+^E.

(21)

(These terms are slightly unfamiliar because they do not arise
in cases of Neumann or Dirichlet boundary conditions.) See Fulling
Fulling:2003zx () for further discussion. That the surface
energy of an interface arises from the volume energy of a smoothed
interface is demonstrated in Milton:2004vy (), and elaborated
in Sect. 2.2.

In the limit of strong coupling, we obtain

limλ→∞E=−π21440a3,

(22)

which is exactly one-half the energy found by Casimir for
perfectly conducting plates casimir ().
Evidently, in this case, the TE modes (calculated here) and
the TM modes (calculated in the following subsection) give equal contributions.

2.1 TM Modes

To verify this last claim, we solve a similar problem with boundary conditions
that the derivative of g is continuous at x=0 and a,

∂∂xg(x,x′)∣∣∣x=0,a is continuous,

(23a)

but the function itself is discontinuous,

g(x,x′)∣∣∣x=a+x=a−=λ∂∂xg(x,x′)∣∣∣x=a,

(23b)

and similarly at x=0. (Here the coupling λ has dimensions of length.)
These boundary conditions reduce, in the limit of strong coupling, to
Neumann boundary conditions on the planes, appropriate to electromagnetic
TM modes:

λ→∞:∂∂xg(x,x′)∣∣∣x=0,a=0.

(23c)

It is completely straightforward to work out the reduced Green’s function
in this case. When both points are between the planes, 0<x,x′<a,

g(x,x′)=12κ\E−κ|x−x′|+12κ~Δ{(λκ2)22coshκ(x−x′)

+λκ2(1+λκ2)[\Eκ(x+x′)+\E−κ(x+x′−2a)]},

(24a)

while if both points are outside the planes, a<x,x′,

g(x,x′)

=

12κ\E−κ|x−x′|

(24b)

+12κ~Δλκ2\E−κ(x+x′−2a)[(1−λκ2)+(1+λκ2)\E2κa],

where the denominator is

~Δ=(1+λκ2)2\E2κa−(λκ2)2.

(25)

It is easy to check that in the strong-coupling limit, the appropriate
Neumann boundary condition (23c) is recovered. For example, in the
interior region, 0<x,x′<a,

limλ→∞g(x,x′)=coshκx<coshκ(x>−a)κsinhκa.

(26)

Now we can compute the pressure on the plane by computing the xx component
of the stress tensor, which is given by (12a),
so we find

txx∣∣x=a−

=

12\I[−κ−2κ~Δ(λκ2)2],

(27a)

txx∣∣x=a+

=

−12\Iκ,

(27b)

and the flux of momentum deposited in the plane x=a is

txx∣∣x=a−−txx∣∣x=a+=\Iκ(2λκ+1)2\E2κa−1,

(28)

and then by integrating over frequency and transverse momentum we obtain
the pressure:

(29)

In the limit of weak coupling, this behaves as follows:

PTM∼−1564π2a6λ2,

(30)

which is to be compared with (15a).
In strong coupling, on the other hand, it has precisely the same limit as
the TE contribution, (15b), which confirms the expectation
given at the end of the previous subsection. Graphs of the two functions
are given in Fig. 2.1.

turn270

TE and TM Casimir pressures between
δ-function planes having strength
λ and separated by a distance a.
In each case, the pressure is plotted as a function
of the dimensionless coupling, λa or λ/a,
respectively, for TE and TM contributions.

For calibration purposes we give the Casimir pressure in practical units
between ideal perfectly conducting parallel plates at zero temperature:

P=−π2240a4ℏc=−1.30 mPa(a/1μm)4.

(31)

2.2 Self-energy of Boundary Layer

Here we show that the divergent self-energy of a single plate,
half the divergent term in (17),
can be interpreted as the energy associated
with the boundary layer. We do this in a simple
context by considering a scalar field
interacting with the background

Lint=−λ2ϕ2σ,

(32)

where the background field σ expands the meaning of the
δ function,

σ(x)={h,−δ2<x<δ2,0,otherwise,

(33)

with the property that hδ=1.
The reduced Green’s function satisfies

[−∂2∂x2+κ2+λσ(x)]g(x,x′)=δ(x−x′).

(34)

This may be easily solved in the region of the slab, −δ2<x<δ2,

g(x,x′)=12κ′{\E−κ′|x−x′|+1^Δ[λhcoshκ′(x+x′)

+(κ′−κ)2\E−κ′δcoshκ′(x−x′)]}.

(35)

Here κ′=√κ2+λh, and

^Δ=2κκ′coshκ′δ+(κ2+κ′2)sinhκ′δ.

(36)

This result may also easily be derived from the multiple reflection
formulas given in Milton:2004ya (), and agrees with that given
by Graham and Olum Graham:2002yr ().

Let us
proceed here with more generality, and consider the stress tensor with
an arbitrary conformal term ccj (),

Tμν=∂μϕ∂νϕ−12gμν(∂λϕ∂λϕ+λhϕ2)−ξ(∂μ∂ν−gμν∂2)ϕ2,

(37)

in d+2 dimensions, d being the number of transverse dimensions,
and ξ is an arbitrary parameter, sometimes called the conformal
parameter.
Applying the corresponding differential operator to the Green’s function
(35), introducing polar coordinates in the (ζ,k) plane,
with ζ=κcosθ, k=κsinθ, and

⟨sin2θ⟩=dd+1,

(38)

we get the following form for the energy density within the slab.

⟨T00⟩

=

2−d−2π−(d+1)/2Γ((d+3)/2)∫∞0\Dκκdκ′^Δ{λh[(1−4ξ)(1+d)κ′2−κ2]cosh2κ′x

(39)

−(κ′−κ)2\E−κ′δκ2},−δ/2<x<δ/2.

We can also calculate the energy density on the other side of the boundary,
from the Green’s function for x,x′<−δ/2,

which vanishes if the conformal value of ξ is used. An identical
contribution comes from the region x>δ/2.

Integrating ⟨T00⟩
over all space gives the vacuum energy of the slab

Eslab

=

−12d+2π(d+1)/2Γ((d+3)/2)∫∞0\Dκκd1κ′^Δ[(κ′−κ)2κ2\E−κ′δδ

(42)

+(λh)2sinhκ′δκ′].

Note that the conformal term does not contribute to the total energy.
If we now take the limit δ→0 and h→∞ so that hδ=1,
we immediately obtain the self-energy of a single δ-function plate:

Eδ=limh→∞Eslab=12d+2π(d+1)/2Γ((d+3)/2)∫∞0\Dκκdλλ+2κ.

(43)

which for d=2 precisely coincides with one-half the constant term in
(17).

There is no surface term in the total Casimir energy as long as the
slab is of finite width, because we may easily check that ddxg∣∣x=x′ is continuous at the boundaries ±δ2. However,
if we only consider the energy internal to the slab we encounter not
only the integrated energy density but a surface term from the integration
by parts—see (21).
It is the complement of this boundary term that gives rise to Eδ,
(43),
in this way of proceeding. That is, as δ→0,

−∫slab(dr)∫dζζ2G(r,r)=0,

(44)

so

Eδ=^E∣∣x=−δ/2+^E∣∣x=δ/2,

(45)

with the normal defining the surface energies pointing into the slab.
This means that in this limit, the slab and surface energies coincide.

Further insight is provided by examining the local energy density.
In this we follow the work of Graham and Olum Graham:2002yr (); Olum:2002ra ().
From (39) we can calculate the behavior of the energy density as the
boundary is approached from the inside:

⟨T00⟩∼Γ(d+1)λh2d+4π(d+1)/2Γ((d+3)/2)1−4ξ(d+1)/d(δ−2|x|)d,|x|→δ/2.

(46)

For d=2 for example, this agrees with the result found in
Graham:2002yr () for ξ=0:

⟨T00⟩∼λh96π2(1−6ξ)(δ/2−|x|)2,|x|→δ2.

(47)

Note that, as we expect, this surface divergence vanishes for the conformal
stress tensor ccj (), where ξ=d/4(d+1). (There will be subleading
divergences if d>2.)
The divergent term in the local energy density from the outside,
(41), as x→−δ/2, is just the negative of that found in
(46). This is why, when the total energy is computed by
integrating the energy density, it is finite for d<2, and independent
of ξ. The divergence encountered for d=2 may be handled by
renormalization of the interaction potential Graham:2002yr ().

Note, further, that for a thin slab, close to the exterior but
such that the slab still appears thin,
x≫δ, the sum of the exterior and interior energy density divergences
combine to give the energy density outside a δ-function potential:

uδ=−λ96π2(1−6ξ)[h(x−δ/2)2−h(x+δ/2)2]=−λ48π21−6ξx3,

(48)

for small x.
Although this limit might be criticized as illegitimate, this result
is correct for a δ-function potential, and
we will see that this divergence structure occurs also in
spherical and cylindrical geometries, so that it is a universal surface
divergence without physical significance, barring gravity.

It is well known as we have just seen that in general the Casimir energy
density diverges in the neighborhood of a surface. For flat surfaces
and conformal theories (such as the conformal scalar theory considered
above Milton:2002vm (), or electromagnetism) those divergences are not
present.2
In particular, Brown and Maclay Brown:1969na ()
calculated the local stress tensor for two ideal plates separated by a distance
a along the z axis, with the result for a conformal scalar

Graham grahamqfext (); Graham:2005cq ()
examined the general relativistic energy conditions
required by causality. In the neighborhood of a smooth domain wall,
given by a hyperbolic tangent, the energy density is always negative
at large enough distances. Thus the weak energy condition is violated, as is
the null energy condition (56). However, when (56) is
integrated over a complete geodesic, positivity is satisfied. It is not clear
if this last condition, the Averaged Null Energy Condition, is always obeyed
in flat space. Certainly it is violated in curved space, but the effects
always seem small, so that exotic effects such as time travel are prohibited.

However, as Deutsch and Candelas deutsch () showed many years ago,
in the neighborhood of a curved surface
for conformally invariant theories, ⟨Tμν⟩ diverges
as ϵ−3, where ϵ is the distance from the surface,
with a coefficient proportional to the sum of the principal curvatures of
the surface. In particular they obtain the result, in the vicinity of
the surface,

⟨Tμν⟩∼ϵ−3T(3)μν+ϵ−2T(2)μν+ϵ−1T(1)μν,

(57)

and obtain explicit expressions for the coefficient tensors T(3)μν
and T(2)μν in terms of the extrinsic curvature of the boundary.

For example, for the case of a sphere, the leading surface divergence has the
form, for conformal fields, for r=a+ϵ, ϵ→0

⟨Tμν⟩=Aϵ3⎛⎜
⎜
⎜
⎜⎝2/a000000000a0000asin2θ⎞⎟
⎟
⎟
⎟⎠,

(58)

in spherical polar coordinates, where the constant is
A=1/720π2 for a scalar field satisfying Dirichlet
boundary conditions, or A=1/60π2
for the electromagnetic field satisfying perfect conductor
boundary conditions.
Note that (58) is properly traceless. The cubic divergence
in the energy density
near the surface translates into the quadratic divergence
in the energy found for a conducting ball miltonballs ().
The corresponding quadratic divergence in the stress corresponds to
the absence of the cubic divergence in ⟨Trr⟩.

This is all completely sensible. However, in their paper Deutsch and Candelas
deutsch () expressed a certain skepticism about the validity of the result
of mildersch () for the spherical shell case (described in part in
Sect. 4.2)
where the divergences cancel. That skepticism was reinforced in a later
paper by Candelas candelas (), who criticized the authors of
mildersch () for omitting δ function terms, and constants
in the energy. These objections seem utterly without merit.
In a later critical paper by the same author
candelas2 (), it was asserted that errors were made, rather than
a conscious removal of unphysical divergences.

Of course, surface curvature divergences are present.
As Candelas noted candelas (); candelas2 (), they have the form

E=ES∫\DS+EC∫\DS(κ1+κ2)+ECI∫\DS(κ1−κ2)2+ECII∫\DSκ1κ2+…,

(59)

where κ1 and κ2 are the principal curvatures of the surface.
The question is
to what extent are they observable. After all, as has been shown
in miltonbook (); Milton:2002vm () and in Sect. 2.2,
we can drastically change the local structure
of the vacuum expectation value of the energy-momentum tensor
in the neighborhood of flat plates by merely
exploiting the ambiguity in the definition of that tensor, yet each
yields the same finite, observable (and observed!) energy of interaction
between the plates. For curved boundaries, much the same is true.
A priori, we do not know which energy-momentum tensor to employ,
and the local vacuum-fluctuation energy density is to a large extent
meaningless. It is the global energy, or the force between distinct bodies,
that has an unambiguous value. It is the belief of the author that divergences
in the energy which go like a power of the cutoff are probably unobservable,
being subsumed in the properties of matter. Moreover, the
coefficients of the divergent terms depend on the regularization scheme.
Logarithmic divergences, of course, are of another class bkv ().
Dramatic cancellations of these curvature terms can occur. It might
be thought that the reason a finite result was found for the Casimir
energy of a perfectly conducting spherical
shell Boyer:1968uf (); balian (); mildersch () is that the term involving the squared difference of curvatures
in (59) is zero only in that case. However,
it has been shown that at least for
the case of electromagnetism the corresponding term
is not present (or has a vanishing coefficient) for an arbitrary smooth
cavity Bernasconi (), and so the Casimir energy for a perfectly conducting
ellipsoid of revolution, for example, is finite.3
This finiteness of the
Casimir energy (usually referred to as the vanishing of the second
heat-kernel coefficient Bordag:2001qi ())
for an ideal smooth closed surface
was anticipated already in balian (),
but contradicted by deutsch (). More specifically, although
odd curvature terms cancel inside and outside for any thin shell, it would
be anticipated that the squared-curvature term, which is present as a
surface divergence in the energy density, would be reflected as an
unremovable divergence in the energy. For a closed surface the last term in
(59)
is a topological invariant, so gives an irrelevant constant,
while no term of the type of the penultimate term can appear due to the
structure of the traced cylinder expansion Fulling:2003zx ().

This section is an adaptation and an extension of calculations presented
in Milton:2004vy (); Milton:2004ya (). This investigation was carried out in
response to the program of the MIT group graham (); graham2 (); Graham:2002fw (); Graham:2003ib (); Weigel:2003tp ().
They first rediscovered irremovable divergences in the Casimir energy
for a circle in 2+1 dimensions first
discovered by Sen sen (); sen2 (), but then found divergences in the case of
a spherical surface, thereby casting doubt on the validity of the
Boyer calculation Boyer:1968uf (). Some of their results, as we shall
see, are spurious, and the rest are well known bkv (). However, their
work has been valuable in sparking new investigations of the problems of
surface energies and divergences.

We now carry out the calculation we presented in Sect. 2
in three spatial dimensions,
with a radially symmetric background

Lint=−12λa2δ(r−a)ϕ2(x),

(60)

which would correspond to a Dirichlet shell in the limit λ→∞.
The scaling of the coupling, which here has dimensions of length,
is demanded by the requirement that the spatial integral of the potential
be independent of a.
The time-Fourier transformed Green’s function satisfies the equation
(κ2=−ω2)

[−∇2+κ2+λa2δ(r−a)]G(r,r′)=δ(r−r′).

(61)

We write G in terms of a reduced Green’s function

G(r,r′)=∑lmgl(r,r′)Ylm(Ω)Y∗lm(Ω′),

(62)

where gl satisfies

[−1r2\D\Drr2\D\Dr+l(l+1)r2+κ2+λa2δ(r−a)]gl(r,r′)=1r2δ(r−r′).

(63)

We solve this in terms of modified Bessel functions, Iν(x), Kν(x),
where ν=l+1/2, which satisfy the Wronskian condition

I′ν(x)Kν(x)−K′ν(x)Iν(x)=1x.

(64)

The solution to (63) is obtained
by requiring continuity of gl at each
singularity, at r′ and a, and the appropriate discontinuity of the
derivative. Inside the sphere we then find (0<r,r′<a)

The same result can be deduced by computing the total energy (16).
The free Green’s function, the first term in (65) or
(68), evidently makes no significant contribution to the energy,
for it gives a term independent of the radius of the sphere, a, so we
omit it. The remaining radial integrals are simply

If we differentiate with respect to a we
immediately recover the force (71). This expression, upon
integration by parts, coincides with that given by Barton barton03 (),
and was first analyzed in detail by Scandurra Scandurra:1998xa ().
This result has also been rederived using the multiple-scattering
formalism Milton:2007wz ().
For strong coupling,
it reduces to the well-known expression for the Casimir energy
of a massless scalar field inside and
outside a sphere upon which Dirichlet boundary conditions are imposed,
that is, that the field must vanish at r=a:

limλ→∞E=−12πa∞∑l=0(2l+1)∫∞0\Dxx\D\Dxln[Iν(x)Kν(x)],

(74)

because multiplying the argument of the logarithm by a power of x is
without effect, corresponding to a contact term. Details of the evaluation
of (74) are given in Milton:2002vm (), and will be
considered in Sect. 4.2 below. (See also
benmil (); lesed1 (); lesed2 ().)

The opposite limit is of interest here. The expansion of the logarithm
is immediate for small λ. The first term, of order λ,
is evidently
divergent, but irrelevant, since that may be removed by renormalization
of the tadpole graph. In contradistinction to the claim of
graham2 (); Graham:2002fw (); Graham:2003ib (); Weigel:2003tp (),
the order λ2 term is finite,
as established in Milton:2002vm (). That term is

E(λ2)=λ24πa3∞∑l=0(2l+1)∫∞0\Dxx\D\Dx[Il+1/2(x)Kl+1/2(x)]2.

(75)

The sum on l can be carried out using a trick due to Klich klich ():
The sum rule